In dealing with refraction of waves, it's important to distinguish between wave fronts and rays and understand how to use both.  The distinction is made at the beginning of section 26-5.  Figure 26-12 in the text shows a construction with both wave fronts and rays and also shows two ways to determine the angles of incidence and refraction.  Here are the two ways.

• The angle of incidence (or refraction) is the angle between the the incident ray (or refracted ray) and the normal to the boundary between the media.

• The angle of incidence (or refraction) is the angle between the the incident wave front (or refracted wave front) and the boundary between the media.

We'll begin with a problem where the wave fronts are parallel to the boundary. Both the angles of incidence and refraction are 0^o in this case. Refer to Figure 1. How do the wave speeds in the media above and below the boundary compare? The key to comparing them is to realize that the constant in this situation is the wave frequency. Remember the problem with the two strings tied together and vibrated at one end with the saw motor? That situation is analogous to this one.  In the case of the strings, the frequency was the same for both strings. That led to the conclusion that wave speed was proportional to wavelength. The same applies to the situation in Figure 1. The wave speed is greater where the wavelength is greater.

Figure 1	Figure 2

Part A

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1. Open this applet. The situation is similar to the one depicted in Figure 1 above. The problem is to determine the frequency, wavelength, and speed of the waves in the media below and above the boundary.

2. Open this applet. The situation is similar to the one depicted in Figure 2 above.  Assume the media are the same as those in Problem 1. That means the wave speeds and wavelengths have the same ratio as in the applet of problem 1. However, the angle of incidence is no longer 0^o. A derivation in section 26-5 of the text shows that v_rsinθ_i = v_isinθ_r. (We're using the subscripts i and r for incident and reflected instead of the 1 and 2 that the book uses.) What is the value of the ratio sinθ_i/sinθ_r? Why will this value be the same as long as the media are the same? Will the ratio be the same if we change the frequency of the incident waves?

3. For the situation of Problem 2, assume that the angle of incidence is 30^o. Determine the angle of refraction to the nearest degree.

4. A photo of water waves passing from deep to shallow water is shown to the right. The waves in the deep water are moving down the screen. The boundary between the regions is shown by a yellow line.  Which of the following statements is correct?

   1. The wave speed is greater in the deeper water, because the wavelength is greater.

   2. The wave speed is greater in the deeper water, because the frequency is smaller.

   3. The wave speed is smaller in the deeper water, because the frequency is smaller.

   4. The wave speed is independent of depth.

Part B

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1. Call the point where wave front 1 intersects the boundary P, and construct the normal to the boundary at this point. Extend the normal on either side of the boundary.

2. Construct the incident ray that intersects the boundary at point P. Then label the angle of incidence as measured with respect to the normal.

3. What is the value of the angle of incidence? You don't need a protractor for this. Use trigonometry.

4. Where the incident waves touch the boundary, construct refracted wave fronts at an angle of 45^o to the boundary. Once again, you don't need a protractor.  Just draw lines with a slope of 1.

5. Construct the refracted ray that intersects the boundary at P. Then label the angle of refraction as measured with respect to the normal.

6. Measure the wavelengths of both the incident and refracted waves.  Indicate on the construction the two distances that you measured. Then use the wavelengths to calculate the ratio of the wave speeds v_r/v_i.

7. Using the angles of incidence and refraction, calculate the ratio of the wave speeds.

8. Compare your answers to steps 6 and 7. Why should they be equal to within measurement error?

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